This paper investigates the study of L-complex fuzzy sets. The L-complex fuzzy set, where L is a completely distributive lattice, is a generalization of the complex fuzzy set. The fundamental set theoretic operations on L-complex fuzzy sets are discussed properly, including L-complex fuzzy complement, union and intersection. New procedures are presented to combine the novel concepts of L-complex fuzzy t-norms and t-conorms and look into the conditions that lead to a comparable representation theorem. We have used the axiomatic method, in the sense that our underlying assumptions, especially about L, are abstract; it can thus be ascertained to what extent our results apply to some new problem. On the other hand, our method shows that if mathematics, as we use it, is consistent, so is fuzziness, as we formulate it.